The lack of long-range structural order in amorphous solids induces well known thermodynamic anomalies, which are the manifestation of distinct peculiarities in the vibrational spectrum. Although the impact of such anomalies vanishes in the long wavelength, elastic continuum limit, it dominates at length scales comparable to interatomic distances, implying an intermediate transition regime still poorly understood. Here we report a study of such mesoscopic domains by means of a broadband version of picosecond photo-acoustics, developed to coherently generate and detect hypersonic sound waves in the sub-THz region with unprecedented sampling efficiency. We identify a temperature-dependent fractal v3/2 frequency behaviour of the sound attenuation, pointing to the presence of marginally stable regions and a transition between the two above mentioned limits. The essential features of this behaviour are captured by a theoretical approach based on random spatial variation of the shear modulus, including anharmonic interactions.